Essex recorded their second win in the competition in 24 hours after the visitors were dismissed for just 134 in pursuit of their opponents' commanding 312 for seven.
Napier claimed four wickets in four balls, having earlier bowled Ricky Ponting for two. Needing to launch their reply with tempo and substance, the Surrey innings floundered with the loss of their first four wickets in 13 overs for 51 runs, with Napier claiming Jason Roy and Ponting.
The former Australia captain had been beaten outside his off stump by Napier but there was no such good fortune for the batsman when he pushed forward to the bowler's next delivery and lost his middle stump, leaving Napier jubilantly celebrating his prized scalp.
The 33-year-old Essex all-rounder continued his demolition job by bowling Azhar Mahmood and Jon Lewis and trapping Zander de Bruyn and Tom Jewell in the crease with full-pitched deliveries.
That left the Surrey innings in tatters at 113 for eight. Napier then found the edge to have Stuart Meaker caught in his next over and complete his collection.
Owais Shah and Ravi Bopara had led the scoring for the home side, with Shah playing a superb innings of 68 from 53 balls including three sixes and four other boundaries. Bopara scored 54 from 58 balls, with five boundaries including a six. Shah and Bopara posted 55 in nine overs for the fourth wicket to initiate the rapid acceleration of the innings.
England all-rounder Bopara was a surprise inclusion for the home side having been given permission to play by the ECB ahead of the Champions Trophy which starts later this week and he responded with a fine and typically stylish half-century containing wristy strokeplay and trademark drives on either side of the wicket until he was caught at deep mid-wicket.
His demise allowed Shah to take over and the former England batsman combined elegant and conventional stroke play with aggressive punishing shots as he dominated the visitors' bowling.
He set the tone for his side who added 191 runs from the final 20 overs, with Shah at the crease for 15 of those. There were briskly struck contributions at the top of the order from Tom Westley and Mark Pettini, who scored 34 and 43 respectively, before James Foster added 39 from 18 deliveries at the conclusion of the innings.